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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-ifp-ncond1 | Structured version Visualization version GIF version |
Description: If one case of an if- condition is false, the other automatically follows. (Contributed by Wolf Lammen, 21-Jul-2024.) |
Ref | Expression |
---|---|
wl-ifp-ncond1 | ⊢ (¬ 𝜓 → (if-(𝜑, 𝜓, 𝜒) ↔ (¬ 𝜑 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ifp 1060 | . 2 ⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))) | |
2 | simpr 489 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝜓) | |
3 | 2 | con3i 157 | . . 3 ⊢ (¬ 𝜓 → ¬ (𝜑 ∧ 𝜓)) |
4 | biorf 935 | . . 3 ⊢ (¬ (𝜑 ∧ 𝜓) → ((¬ 𝜑 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (¬ 𝜓 → ((¬ 𝜑 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))) |
6 | 1, 5 | bitr4id 294 | 1 ⊢ (¬ 𝜓 → (if-(𝜑, 𝜓, 𝜒) ↔ (¬ 𝜑 ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 845 if-wif 1059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-ifp 1060 |
This theorem is referenced by: wl-ifp-ncond2 35147 |
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